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Theorem tfi 4248
Description: The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring] p. 39. This principle states that if is a class of ordinal numbers with the property that every ordinal number included in also belongs to , then every ordinal number is in .

(Contributed by NM, 18-Feb-2004.)

Assertion
Ref Expression
tfi  C_  On  On  C_  On
Distinct variable group:   ,

Proof of Theorem tfi
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-ral 2305 . . . . . . 7  On  C_  On  C_
2 imdi 239 . . . . . . . 8  On  C_  On  C_  On
32albii 1356 . . . . . . 7  On  C_  On  C_  On
41, 3bitri 173 . . . . . 6  On  C_  On  C_  On
5 dfss2 2928 . . . . . . . . . 10 
C_
65imbi2i 215 . . . . . . . . 9  On 
C_  On
7 19.21v 1750 . . . . . . . . 9  On  On
86, 7bitr4i 176 . . . . . . . 8  On 
C_  On
98imbi1i 227 . . . . . . 7  On  C_  On  On  On
109albii 1356 . . . . . 6  On  C_  On  On  On
114, 10bitri 173 . . . . 5  On  C_  On  On
12 simpl 102 . . . . . . . . . . 11  On
13 tron 4085 . . . . . . . . . . . . . 14  Tr  On
14 dftr2 3847 . . . . . . . . . . . . . 14  Tr  On  On  On
1513, 14mpbi 133 . . . . . . . . . . . . 13  On  On
1615spi 1426 . . . . . . . . . . . 12  On  On
1716spi 1426 . . . . . . . . . . 11  On  On
1812, 17jca 290 . . . . . . . . . 10  On  On
1918imim1i 54 . . . . . . . . 9  On  On
20 impexp 250 . . . . . . . . 9  On  On
21 impexp 250 . . . . . . . . . 10  On  On
22 bi2.04 237 . . . . . . . . . 10  On  On
2321, 22bitri 173 . . . . . . . . 9  On  On
2419, 20, 233imtr3i 189 . . . . . . . 8  On  On
2524alimi 1341 . . . . . . 7  On  On
2625imim1i 54 . . . . . 6  On  On  On  On
2726alimi 1341 . . . . 5  On  On  On  On
2811, 27sylbi 114 . . . 4  On  C_  On  On
2928adantl 262 . . 3  C_  On  On  C_  On  On
30 sbim 1824 . . . . . . . . . 10  On  On
31 clelsb3 2139 . . . . . . . . . . 11  On  On
32 clelsb3 2139 . . . . . . . . . . 11
3331, 32imbi12i 228 . . . . . . . . . 10  On  On
3430, 33bitri 173 . . . . . . . . 9  On  On
3534ralbii 2324 . . . . . . . 8  On  On
36 df-ral 2305 . . . . . . . 8  On  On
3735, 36bitri 173 . . . . . . 7  On  On
3837imbi1i 227 . . . . . 6  On  On  On  On
3938albii 1356 . . . . 5  On  On  On  On
40 ax-setind 4220 . . . . 5  On  On  On
4139, 40sylbir 125 . . . 4  On  On  On
42 dfss2 2928 . . . 4  On  C_  On
4341, 42sylibr 137 . . 3  On  On 
On  C_
4429, 43syl 14 . 2  C_  On  On  C_ 
On  C_
45 eqss 2954 . . 3  On 
C_  On  On  C_
4645biimpri 124 . 2  C_  On  On  C_  On
4744, 46syldan 266 1  C_  On  On  C_  On
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97  wal 1240   wceq 1242   wcel 1390  wsb 1642  wral 2300    C_ wss 2911   Tr wtr 3845   Oncon0 4066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071
This theorem is referenced by:  tfis  4249
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